Let's say there's a 2Ah battery with a C rating of 2C.
The above statement is incorrect.
The correct statement would be 'Let's say there's a C2 rated battery having a capacity of 2 Ah'. It would refer to a 2 Ah battery capable of delivering 1 A maximum for 2 hours without getting damaged.
Consider another example of a C20 rated battery having a capacity of 100 Ah. It would refer to a 100 Ah battery capable of delivering 5 A maximum for 20 hours without getting damaged.
It should be borne in mind that the rated capacity would not be fully available at higher discharge rates.
0.1 C or 0.5 C or 1 C refer to charge or discharge current. For example, charging a 100 Ah battery at 0.1 C would mean a charging current of 100 x 0.1 = 10A. Likewise for discharge current.
How do can I increase the current discharge for this particular battery?
The discharge current of a particular battery would be dependant on its voltage and the resistance of the load resistance.